| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Single locus sketching |
| Difficulty | Moderate -0.8 This question tests basic complex number operations: finding modulus (straightforward calculation), finding argument of conjugate (routine), solving a linear equation with conjugates (algebraic manipulation), and sketching a perpendicular bisector locus (standard FP1 result). All parts are textbook exercises requiring recall and direct application of formulas with no problem-solving insight needed. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(\sqrt{13}\) | B1 | Obtain correct answer, decimals OK |
| (ii) \(\approx -0.59\) | M1 A1 A1 | Using \(\tan^{-1}\frac{b}{a}\), or equivalent trig allow \(\pm\) or \(-\); Obtain 0.59; Obtain correct answer |
| (b) \(1 - 2i\) | M1 A1A1 A1 | Express LHS in Cartesian form & equate real and imaginary parts; Obtain \(x = 1\) and \(y = -2\); Correct answer written as a complex number |
| (c) Sketch of vertical straight line through \((-0.5, 0)\) | B1 B1 | Sketch of vertical straight line; Through \((-0.5, 0)\) |
(a) (i) $\sqrt{13}$ | B1 | Obtain correct answer, decimals OK
(ii) $\approx -0.59$ | M1 A1 A1 | Using $\tan^{-1}\frac{b}{a}$, or equivalent trig allow $\pm$ or $-$; Obtain 0.59; Obtain correct answer
(b) $1 - 2i$ | M1 A1A1 A1 | Express LHS in Cartesian form & equate real and imaginary parts; Obtain $x = 1$ and $y = -2$; Correct answer written as a complex number
(c) Sketch of vertical straight line through $(-0.5, 0)$ | B1 B1 | Sketch of vertical straight line; Through $(-0.5, 0)$
**Total: 10 marks**7
\begin{enumerate}[label=(\alph*)]
\item The complex number $3 + 2 \mathrm { i }$ is denoted by $w$ and the complex conjugate of $w$ is denoted by $w ^ { * }$. Find
\begin{enumerate}[label=(\roman*)]
\item the modulus of $w$,
\item the argument of $w ^ { * }$, giving your answer in radians, correct to 2 decimal places.
\end{enumerate}\item Find the complex number $u$ given that $u + 2 u ^ { * } = 3 + 2 \mathrm { i }$.
\item Sketch, on an Argand diagram, the locus given by $| z + 1 | = | z |$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2006 Q7 [10]}}